Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.

Motivation: I plan to use this list in my teaching, to motivate general education undergraduates, and early year majors, suggesting to them an idea of what research mathematicians do.

Meaning of "not too famous" Examples of problems that are too famous might be the Goldbach conjecture, the $3x+1$-problem, the twin-prime conjecture, or the chromatic number of the unit-distance graph on ${\Bbb R}^2$. Roughly, if there exists a whole monograph already dedicated to the problem (or narrow circle of problems), no need to mention it again here. I'm looking for problems that, with high probability, a mathematician working outside the particular area has never encountered.

Meaning of: anyone can understand The statement (in some appropriate, but reasonably terse formulation) shouldn't involve concepts beyond (American) K-12 mathematics. For example, if it weren't already too famous, I would say that the conjecture that "finite projective planes have prime power order" does have barely acceptable articulations.

Meaning of: long open The problem should occur in the literature or have a solid history as folklore. So I do not mean to call here for the invention of new problems or to collect everybody's laundry list of private-research-impeding unproved elementary technical lemmas. There should already exist at least of small community of mathematicians who will care if one of these problems gets solved.

I hope I have reduced subjectivity to a minimum, but I can't eliminate all fuzziness -- so if in doubt please don't hesitate to post!

To get started, here's a problem that I only learned of recently and that I've actually enjoyed describing to general education students.

You might get more success if you sampled certain open problem lists and indicated which ones fit your list and which ones did not. I could mention various combinatorial problems such as integer complexity, determinant spectrum, covering design optimization, but I can't tell from your description if they would be suitable for you. Gerhard "They Are Suitable For Me" Paseman, 2012.06.21
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Gerhard PasemanJun 21 '12 at 19:11

To save the search for explanation of cryptic acronyms for those of us outside US, K-12 means high school. @Mahmud: You are using a wrong meaning of the word “problem”. The TSP is not an unproved mathematical statement, it is a computational task.
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Emil JeřábekJun 22 '12 at 12:05

97 Answers
97

One problem which I think is mentioned in Guy's book is the integer block problem: does there exist a cuboid (aka "brick") where the width, height, breadth, length of diagonals on each face, and the length of the main diagonal are all integers?

update 2012-07-12 Since the question has returned to the front page, I'm taking the liberty to add some links that I found after Scott Carnahan's comments. (Scott deserves the credit, really, but I thought the links belonged in the answer rather than in the comments.)

Because so much has been known about Pythagorean triples for so long, I'm shocked that this problem is open. Is there an intuitive explanation of why the problem is so hard?
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VectornautJun 22 '12 at 5:38

The solution space forms an algebraic surface in the projectivized space of box dimensions. The surface has rather high degree, and in fact van Luijk showed that it is of general type, (and therefore rather resistant to standard methods).
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S. Carnahan♦Jun 22 '12 at 6:03

1

Arguably, the brick violates the "outside mathematician" condition. I even tried to convince some crank (may one say this word here? :-) not to waste as many of his lifetime to the problem as I did.
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Hauke ReddmannMay 14 at 14:49

The Casas-Alvero conjecture: let the characteristic of the field $k$ be $0$. If a monic polynomial $f\in k[X]$ of degree $n$ has a common root with each of its derivatives $f',\ldots,f^{(n-1)}$, then $f(X)=(X-a)^n$ for some $a\in k$.

@Joel. Right! If $k$ is of finite characteristic $p$, then $X^{2p}+X^p$ does share a root with every derivative, but is not a monomial.
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Denis SerreJun 22 '12 at 20:38

10

For those interested in this conjecture, here is what I believe the current state of knowledge on the conjecture : arxiv.org/abs/math/0605090 The first open case is $n=12$. Interestingly, the proofs in the known cases use scheme theory (over $\mathbf{Z}$).
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François BrunaultJun 22 '12 at 22:57

5

@François Brunault. Some months ago I asked this question mathoverflow.net/questions/94838/… with the Casas-Alvero conjecture in mind. It appeared from the answers that instead of the argument using scheme theory, the simpler Lefschetz principle ( proofwiki.org/wiki/Lefschetz_Principle_(First-Order) ) can be used. (answering to my question, Qiaochu Yuan also indicated an ultraproduct construction which is even simpler than the Lefschetz Principle, since no completeness result is used).
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js21Jun 25 '12 at 6:22

Consider $k + 1$ runners on a circular track of unit length. At $t = 0$, all runners are at the same position and start to run; the runners' speeds are pairwise distinct. A runner is said to be lonely if at distance of at least $1/(k + 1)$ from each other runner. The lonely runner conjecture states that every runner gets lonely at some time.

Also, I suspect this is equivalent to the lonely starting post conjecture, which is the conjecture above except that one of the runners has speed 0 and the statement is that he/she gets lonely. Gerhard "Ask Me About Going Slow" Paseman, 2012.06.22
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Gerhard PasemanJun 22 '12 at 18:59

57

Observe that the human condition implies that everyone gets lonely at some time. In particular the runners get lonely.
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Asaf KaragilaJun 22 '12 at 20:59

39

This is open for $k\geq 7$. The proof for $k=6$ was done by Barajas and Serra using elaborate computer-assisted casework, and many simplifications that rely on the fact that $6+1$ is prime. It is worth noting that when the ratio of two speeds is irrational, the problem is made easier by density arguments, so the essentially hardest case is when all the speeds are integers. Therefore this is a combinatorial number theory question disguised as basic calculus.
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Andrew DudzikJul 2 '12 at 2:14

This is the second time I've seen this question on mathoverflow and this will be the second time I'vve posted this answer.

Singmaster's conjecture says there is a finite upper bound on the number of times a number (other than the $1$s on the edge) can appear in Pascal's triangle. The upper bound may be as low as $8$. If so, then no number (besides those $1$s) appears more than eight times in Pascal's triangle. Only one number is known to appear that many times:
$$
\binom{3003}{1} = \binom{78}{2} = \binom{15}{5} = \binom{14}{6}
$$

It has been proved that infinitely many numbers appear twice; similarly three times, four times, and six times. It is unknown whether any number appears five times or seven times.

Singmaster states that Erdős said the conjecture is probably true but probably difficult to prove.

We don't really need Erdős to tell us it's probably true when we can do straightforward probabilistic estimates (plus some geometry of plane curves). A short computation shows that there are no numbers less than $10^{1000}$ that have odd multiplicity greater than 3, and heuristics suggest it is quite unlikely that such numbers exist.
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S. Carnahan♦Jul 2 '12 at 9:49

3

@S.Carnahan : How did you do that "short computation"?
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Michael HardyJul 6 '12 at 21:49

2

Odd multiplicity means you have a number of the form $\binom{2k}{k}$. It's not hard to check whether a number has the form of a binomial coefficient $\binom{m}{n}$ in SAGE, since you have a built-in function that estimates integer $n$-th roots.
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S. Carnahan♦Jul 12 '12 at 7:02

7

I love this problem! Everything about it is simple and compelling, and it can be understood by anyone who knows how to add. Is there also a simple heuristic argument for why it should be true? @S. Carnahan , can you flesh out your heuristics a little more? What's this stuff about geometry of plane curves?
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VectornautJul 22 '12 at 18:44

wow, at first look it seems hard to believe that this is still a conjecture!
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SuvritJun 22 '12 at 3:26

15

As I understand it, this kind of identity is amenable in principle to automatic theorem-proving methods, but (using known techniques) is out of reach of current computers.
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Timothy ChowJun 22 '12 at 14:40

4

Tim, there is also an example, from December 2011, for $1/\pi^4$ due to Jim Cullen (members.bex.net/jtcullen515), another mathematics amateur; I cannot easily fine it online though.
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Wadim ZudilinAug 25 '12 at 11:13

Do all elements of this sequence with large enough index $n$ lie in the interval $(0,1/2)$?

It is known that $\beta^n$ is uniformly distributed modulo one for almost all $\beta>1$, but explicit examples of $\beta$ for which density holds are not known. This question seems to originate in work of Weyl and Koksma on uniform distribution.

Update: Since posting this answer I've attempted to find some references with which to flesh it out, with only modest success. The earlier paper I have identified which deals with this question directly is T. Vijayaraghavan's 1940 article On the fractional parts of the powers of a number, in which it is shown that the sequence $(3/2)^n \mod 1$ has infinitely many limit points. Mahler conjectured in 1968 that the answer to his question is negative. Jeffrey Lagarias' 1985 survey on the Collatz problem, The 3x + 1 Problem and Its Generalizations, includes a one-page overview of the literature on the distribution of this sequence. Flatto, Lagarias and Pollington subsequently proved that the diameter of the set of accumulation points is at least 1/3; Mahler's question would be answered in the negative if this is improved to "at least 1/2".

There is a lot of number theory elementary conjectures, but one that is especially elementary is the so called Giuga Conjecture (or Agoh-Giuga Conjecture), from the 1950:
a positive integer $p>1$ is prime if and only if
$$\sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod{p}$$

I'm not so sure. Which popular books, widespread textbooks or articles do you know where it is stated?
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Georges ElencwajgJun 22 '12 at 17:12

8

Popular books, I don't know, but David Feldman wrote, "I'm looking for problems that, with high probability, a mathematician working outside the particular area has never encountered." (Emphasis mine.) It feels to me that most professional mathematicians, even those not working in transcendental number theory, are familiar with this one.
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Timothy ChowJun 22 '12 at 18:17

Additional info: The best known result is that all triangles of maximum angle 100 degrees admit a periodic orbit. It is also known that all triangles (in fact, all polygons) with angles that are rational multiples of $\pi$ admit periodic orbits.
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Alex BeckerJul 3 '12 at 4:08

I thought this might be a good candidate since that book was meant as a bridge from competitive Mathematics to research. There are a few other examples, but I am quoting only one here due to your requirement.

But I feel like this is not a good introduction to what actual research, at least for a beginning researcher, is like. Usually, you are taught fairly advanced methods and some result that was achieved using those methods and then are asked to modify it a little bit to see what you can do.
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David CorwinJun 24 '12 at 18:30

Problem: The partition function $p(n)$ is even (resp. odd) half of the time.

Of course you need to explain to a general audience what the partition function is, but that's not hard, my daughter in K1 got as an assignment to compute $p(n)$ for $n$ up to 4.
You also need to explain "half of the time", which means that the number of $n < x$ such that $p(n)$ is even, divided by $x$, has limit 1/2 when $x$ goes to infinity, so you need the notion of limit of a sequence, which is in K12, isn't it ?

The problem is certainly famous among specialists, but not too famous. I don't think there are books on it, for instance. It is old (formulated as a conjecture during the 50th), with an history going back to Ramanajunan. And I like it very much.

The notion of limit of a sequence is not usually taught in the US until a real analysis course, which is usually taken only by students in mathematics and frequently not until the third (or even last) year of university. (But I think this case is concrete enough that the necessary ideas here could be explained to a high school student.)
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Alexander WooJun 22 '12 at 4:05

4

Sequences are taught before real analysis, usually in Calc 2 along with infinite series. And the more basic material is suitable for high school, even a decent precalculus class. These are only sequences of reals so it isn't very general, and while they are taught, students might not really "understand" them until later.
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Francis AdamsJun 22 '12 at 12:51

1

Yes, there is an option for seniors in a good high school to learn some calculus, but most calculus courses in the United States no longer give a rigourous definition of a limit. Without a rigourous definition, there are some subtle possibilities for what might go wrong that won't be appreciated. (Of course, very few students at that level have the mathematical maturity to understand a rigourous definition well enough to appreciate the subtle possibilities anyway, which is why the rigourous definition isn't taught anymore.)
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Alexander WooSep 6 '12 at 4:11

2

Also, "half of the time" can be restated in probabilistic terms. In other words, instead of framing it as a real analysis question, appeal to probabilistic intuition. Alexander Woo's remarks about subtle possibilities notwithstanding, vastly larger numbers of students learn elementary probability and statistics than calculus.
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Victor ProtsakJan 6 '14 at 19:28

@unknown: That's a fair comment. Still, if the goal is to find conjectures that are accessible to the general math-loving public that they may not have heard of before, I think the decimal expansion problem counts. Perhaps David Feldman can clarify whether he really means that 90% of non-number theorists haven't heard of the conjecture of which this happens to be a corollary, or whether he means something weaker than that.
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Timothy ChowJun 22 '12 at 14:37

The circulant Hadamard matrix conjecture, first stated in print by Ryser in 1963. It can be stated as follows. If $n>4$, then there does not exist a sequence $(a_1,a_2,\dots,a_n)$ of $\pm 1$'s satisfying
$$ \sum_{i=1}^n a_i a_{i+k}=0,\ 1\leq k\leq n-1, $$
where the subscript $i+k$ is taken modulo $n$.

Further related: Let m be the largest integer such that the integer interval (-m,m) is contained in the set D_n, the set of determinants of order n 0-1 matrices. What function of n are very good bounds for approximating m? Cf determinant spectrum on Will Orrick's maxdet site. Gerhard "Ask Me About Binary Matrices" Paseman, 2012.06.22
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Gerhard PasemanJun 22 '12 at 19:08

$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan(t)+\sqrt{7}}{\tan(t)-\sqrt{7}}\right|\ dt = \sum_{n\geq
1} \left(\frac n7\right)\frac{1}{n^2}, $$
where $\displaystyle\left(\frac n7\right)$ denotes the Legendre symbol. Not really
my favorite identity, but it has the interesting feature that it is a
conjecture! It is a rare example of a conjectured explicit identity
between real numbers that can be checked to arbitrary accuracy.
This identity has been verified to over 20,000 decimal places.
See J. M. Borwein and D. H. Bailey, Mathematics by Experiment:
Plausible Reasoning in the 21st Century, A K Peters, Natick, MA,
2004 (pages 90-91).

It was a good idea to split the two conjectures to two answers, but you should have done it the other way around. I venture to guess that most people, like me, originally upvoted this answer because of Sendov’s conjecture, not because of an obscure integral equality which I couldn’t explain to any high school student I now of.
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Emil JeřábekJun 25 '12 at 10:34

1

@Emil: Emil, The answers were split because of an user requesting me to do so. Otherwise I would have kept it here itself.
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S.C.Jul 1 '12 at 7:40

Does every (Jordan) curve in the plane contain all four vertices of
some square?

Update: Here is a variation due to Helge Tverberg: Does every (polygonal) curve in the plane outside of the unit circle, contain all four vertices of some square with side length >0.1? This version implies the original problem and lacks disadvantages pointed out by Tim Chow and Henry Cohn.

This is a nice problem but it's only open in the case where the curve is pathologically ugly, in a way that perhaps not "anyone can understand."
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Timothy ChowJun 22 '12 at 2:05

5

I do think that anyone can understand whats an injective, continuous map from the circle to the plane.
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Fernando MuroJun 22 '12 at 6:44

31

Actually, I disagree that anyone can (quickly, easily) understand what such a map is for the purposes of this problem, since the maps for which it's not known are of a sort even mathematicians didn't realize existed until well into the 19th century. One can still state the problem, but it's likely to lead to conversations of the following sort. "Wow, so you mean nobody knows in advance if this curve [draws a curve] has a square in it?" "Well, actually we know that case, or really any curve you can draw, but mathematicians have discovered exotic curves for which we don't know the answer."
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Henry CohnJun 22 '12 at 13:14

15

The issue here is that intuitive "definitions" of continuous tend to be wrong. "You can draw it without lifting your pencil" really means at least piecewise smooth.
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Noah SnyderJun 24 '12 at 3:40

Let $H_n=\sum_{j=1}^n 1/j$. Then for all $n\geq 1$,
$$ \sum_{d|n}d\leq H_n+(\log H_n)e^{H_n}. $$
Jeff Lagarias showed that this is equivalent to the Riemann hypothesis!

Let $x_0=2$, $x_{n+1}=x_n-\frac{1}{x_n}$ for $n\geq 0$. Then $x_n$ is unbounded.

The largest integer that cannot be written in the form $xy+xz+yz$, where $x,y,z$ are positive integers, is 462. It is known that there exists at most one such integer $n>462$, which must be greater than $2\cdot 10^{11}$. See J. Borwein and K.-K. S. Choi, On the
representations of $xy+yz+xz$, Experiment. Math.9 (2000), 153-158; http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.em/1046889597.

I'm wondering if I "get" #2. I see an implicit map from $S^1$ to $S^1$ of index 2, so yes, it seems generally hard to understand the dynamical fate of a given starting value. A similar question might ask if the binary expansion of $\sqrt{2}$ contains strings of 0's of arbitrary length. But is #2 specifically conjugate to something more familiar?
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David FeldmanJun 23 '12 at 19:17

1

@Davidac897: I think the conjecture part of #3 is the first sentence: "The largest integer... is 462." If I'm reading the rest correctly, it's known that if the conjecture is false, it's only because of a single counterexample that must be greater than 200 billion.
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Owen BieselJul 27 '12 at 21:15

3

Question #2 was addressed in the paper math.grinnell.edu/~chamberl/papers/mario_digits.pdf The real problem concerns the initial value $x_0=2$. It can be shown that the set of initial values which produce an unbounded sequence $\{x_n\}$ has full measure, so from a probabilistic perspective, one expects the statement in question 2 to hold.
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Marc ChamberlandAug 19 '12 at 18:40

For a polynomial $$f(z) = (z-r_{1}) \cdot (z-r_{2}) \cdots (z-r_{n}) \quad \text{for} \ \ \ \ n \geq 2$$ with all roots $r_{1}, ..., r_{n}$ inside the closed unit disk $|z| \leq 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point of $f$.

Remarks: Although Catalan's constant is certainly well-known, the irrationality is
the tip of the iceberg of a related conjecture of Milnor about the linear independence
over the rationals of volumes of certain hyperbolic 3-manifolds (which is a special
case of a conjecture of Ramakrishnan). The
irrationality of Catalan's constant would imply that the volume of the
unique hyperbolic structure on the Whitehead link complement is irrational.
To this date, it is not known that any hyperbolic 3-manifold has irrational
volume.

I've known ring theory for a while, and it never even occurred to me that that was difficult (let alone possibly true).
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David CorwinJul 22 '12 at 20:35

9

Mel, to whom I will be eternally grateful for my low Erdos number and much else, was a master of the uniquely mathematical game of one downsmanship: "You don't if ..., we I don't even know if ...!!!
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David FeldmanJul 23 '12 at 0:04

The Kneser–Poulsen conjecture in dimension 3: An arrangement of (possibly overlapping) unit balls in space is tighter than a second arrangement of the same balls if, for all $i$ and $j$, the distance between the centers of ball $i$ and ball $j$ in the first arrangement is less than or equal to the distance between the centers of ball $i$ and ball $j$ in the second arrangement. The conjecture is that a tighter arrangement always has equal or smaller total volume. True in the plane, open in higher dimensions.

Here is another easy to state problem which is 140 years old but not very famous.
Consider the potential of finitely many positive charges:
$$u(x)=\sum_{j=1}^n\frac{a_j}{|x-x_j|},\quad x,x_j\in R^3,\quad a_j>0$$
How many equilibrium points can this potential have? Equilibrium points are solutions
of $\nabla u(x)=0$.

First conjecture: it is always finite.

Second conjecture: when finite, it is at most $(n-1)^2$. This estimate is stated by Maxwell
in his Treatease on Electricity and Magnetism, vol. I, section 113, as something known.
The editor
(J. J. Thomson) wrote a footnote that he "could not find any place where this result is proved".

Nobody could find this place to this time. This is even unknown in the simplest case
when all $a_j=1$ and $n=3$.

Melvyn Nathanson, in his book Elementary Methods in Number Theory (Chapter 8: Prime Numbers) states the following:

A conjecture of Schinzel and Sierpinski asserts that every positive rational number $x$ can be represented as a quotient of shifted primes, that $x=\frac{p+1}{q+1}$ for primes $p$ and $q$. It is known that the set of shifted primes, generates a subgroup of the multiplicative group of rational numbers of index at most $3$.

Erdos's problem on the length of lemniscates (it is somewhat famous in certain narrow circles).
Let $P$ be a polynomial, and consider the set $E=\{ z:|P(z)|=1\}$ in the complex plane.

What is the maximum length of $E$ over all monic polynomials of degree $d$?

Erdos conjectured that an extremal $P$ is $P_0(z)=z^d+1$.

It is known that the asymptotic of maximal length is $2d+o(d).$
It is known that $P_0$ gives a local maximum. It is also known that for every
extremal polynomial,
all critical points lie on $E$, so $E$ must be connected.

However the conjecture is not established even for $d=3$.

After Erdos's death, I offered a $200 prize for the first solution. (Erdos had offered the same, but I do not know whether one can collect his prize.)

That's presumably the intention, though the problem as stated looks simpler... (The "congrent number problem" amounts to asking which integers are the areas of right triangles all of whose sides are rational.)
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Noam D. ElkiesJun 23 '12 at 3:09

The Cerny conjecture says that if X is a collection of mappings on an n element set such that some iterated composition (repetitions allowed) of elements of X is a constant map then there is a composition of at most $(n-1)^2$ mappings from X which is a constant mapping. This comes from automata theory. See http://en.m.wikipedia.org/wiki/Synchronizing_word.